Optical fiber with a core ring

ABSTRACT

An optical fiber includes a cladding with a material having a first refractive index and a pattern of regions formed therein. Each of the regions has a second refractive index lower than the first refractive index. The optical fiber further includes a core region and a core ring having an inner perimeter, an outer perimeter, and a thickness between the inner perimeter and the outer perimeter. The thickness is sized to reduce the number of ring surface modes supported by the core ring.

RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 11/971,181, filed Jan. 8, 2008 and incorporated in its entiretyby reference herein, which is a continuation of U.S. patent applicationSer. No. 11/737,683, filed Apr. 19, 2007 (now U.S. Pat. No. 7,400,806,issued Jul. 15, 2008), and incorporated in its entirety by referenceherein, which is a continuation of U.S. patent application Ser. No.11/123,879, filed May 6, 2005 (now U.S. Pat. No. 7,228,041, issued Jun.5, 2007) and incorporated in its entirety by reference herein, which isa continuation-in-part of U.S. patent application Ser. No. 10/938,755,filed Sep. 10, 2004 (now U.S. Pat. No. 7,110,650, issued on Sep. 19,2006) and incorporated in its entirety by reference herein, and claimsthe benefit of priority under 35 U.S.C. § 119(e) of U.S. ProvisionalApplication Nos. 60/502,329, 60/502,390, and 60/502,531, each filed onSep. 12, 2003, and of U.S. Provisional Application No. 60/564,896, filedApr. 23, 2004. U.S. patent application Ser. No. 11/123,879 claims thebenefit of priority under 35 U.S.C. § 119(e) of U.S. ProvisionalApplication No. 60/569,271, filed on May 8, 2004 and incorporated in itsentirety by reference herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present application is in the field of optical fibers forpropagating light, and more particularly is in the field ofphotonic-bandgap fibers having a hollow core, or a core with arefractive index lower than the cladding materials.

2. Description of the Related Art

Photonic-bandgap fibers (PBFs) have attracted great interest in recentyears due to their unique advantages over conventional fibers. Inparticular, the propagation loss in an air-core PBF is not limited bythe core material, and it is expected that the propagation loss can beexceedingly low. The nonlinear effects in an air-core PBF are verysmall, and in certain PBFs, the core can be filled with liquids or gasesto generate the desired light-matter interaction. Numerous newapplications enabled by these advantages have been demonstratedrecently. Such applications are described, for example, in BurakTemelkuran et al., Wavelength-scalable hollow optical fibres with largephotonic bandgaps for CO ₂ laser transmission, Nature, Vol. 420, 12 Dec.2002, pages 650-653; Dimitri G. Ouzounov et al., Dispersion andnonlinear propagation in air-core photonic band-gap fibers, Proceedingsof Conference on Laser and Electro-Optics (CLEO) 2003, Baltimore, USA,1-6 Jun. 2003, paper CThV5, 2 pages; M. J. Renn et al., Laser-GuidedAtoms in Hollow-Core Optical Fibers, Physical Review Letters, Vol. 75,No. 18, 30 Oct. 1995, pages 3253-3256; F. Benabid et al., Particlelevitation and guidance in hollow-core photonic crystal fiber, OpticsExpress, Vol. 10, No. 21, 21 Oct. 2002, pages 1195-1203; and KazunoriSuzuki et al., Ultrabroad band white light generation from a multimodephotonic bandgap fiber with an air core, Proceedings of Conference onLaser and Electro-Optics (CLEO) 2001, paper WIPD1-11, pages 24-25, whichare hereby incorporated herein by reference.

Calculations of selected properties of the fundamental mode of the PBFshave also been reported in, for example, R. F. Cregan et al.,Single-Mode Photonic Band Gap Guidance of Light in Air, Science, Vol.285, 3 Sep. 1999, pages 1537-1539; Jes Broeng et al., Analysis of airguiding photonic bandgap fibers, Optics Letters, Vol. 25, No. 2, Jan.15, 2000, pages 96-98; and Jes Broeng et al., Photonic Crystal Fibers: ANew Class of Optical Waveguides, Optical Fiber Technology, Vol. 5, 1999,pages 305-330, which are hereby incorporated herein by reference.

Surface modes, which do not exist in conventional fibers, are defectmodes that form at the boundary between the air core and thephotonic-crystal cladding. Surface modes can occur when an infinitephotonic crystal is abruptly terminated, as happens for example at theedges of a crystal of finite dimensions. Terminations introduce a newset of boundary conditions, which result in the creation of surfacemodes that satisfy these conditions and are localized at thetermination. See, for example, F. Ramos-Mendieta et al., Surfaceelectromagnetic waves in two-dimensional photonic crystals: effect ofthe position of the surface plane, Physical Review B, Vol. 59, No. 23,June 1999, pages 15112-15120, which is hereby incorporated herein byreference.

In a photonic crystal, the existence of surface modes depends stronglyon the location of the termination. See, for example, A. Yariv et al.,Optical Waves in Crystals: Propagation and Control of Laser Radiation,John Wiley & Sons, New York, 1984, pages 209-214, particularly at page210; and J. D. Joannopoulos et al., Photonic Crystals. Molding the flowof light, Princeton University Press, Princeton, N.J., 1995, pages54-77, particularly at page 73; which are hereby incorporated herein byreference; and also see, for example, F. Ramos-Mendieta et al., Surfaceelectromagnetic waves in two-dimensional photonic crystals: effect ofthe position of the surface plane, cited above. For example, in photoniccrystals made of dielectric rods in air, surface modes are induced onlywhen the termination cuts through rods. A termination that cuts onlythrough air is too weak to induce surface modes. See, for example, J. D.Joannopoulos et al., Photonic Crystals: Molding the flow of light, citedabove.

Unless suitably designed, a fiber will support many surface modes.Recent demonstrations have shown that surface modes play a particularlyimportant role in air-core PBFs, and mounting evidence indicates thatsurface modes impose serious limitations in air-core photonic-bandgapfibers by contributing to propagation losses. See, for example, K.Saitoh et al., Air-core photonic band-gap fibers: the impact of surfacemodes, Optics Express, Vol. 12, No. 3, February 2004, pages 394-400;Douglas C. Allan et al., Surface modes and loss in air-core photonicband-gap fibers, in Photonic Crystals Materials and Devices, A. Adibi etal. (eds.), Proceedings of SPIE, Vol. 5000, 2003, pages 161-174; WahTung Lau et al., Creating large bandwidth line defects by embeddingdielectric waveguides into photonic crystal slabs, Applied PhysicsLetters, Vol. 81, No. 21, 18 Nov. 2002, pages 3915-3917; Dirk Müller etal., Measurement of Photonic Band-gap Fiber Transmission from 1.0 to 3.0μm and impact of Surface Mode Coupling, Proceedings of Conference onLaser and Electro-Optics (CLEO) 2003, Baltimore, USA, 1-6 Jun. 2003,paper QTuL2, 2 pages; Hyang Kyun Kim et al., Designing air-corephotonic-bandgap fibers free of surface modes, IEEE Journal of QuantumElectronics, Vol. 40, No. 5, May 2004, pages 551-556; and Michel J. F.Digonnet et al., Simple geometric criterion to predict the existence ofsurface modes in air-core photonic-bandgap fibers, Optics Express, Vol.12, No. 9, May 2004, pages 1864-1872, which are hereby incorporatedherein by reference. Also see, for example, J. D. Joannopoulos et al.,Photonic Crystals: Molding the flow of light, cited above; A. Yariv etal., Optical Waves in Crystals: Propagation and Control of LaserRadiation, cited above; and F. Ramos-Mendieta et al., Surfaceelectromagnetic waves in two-dimensional photonic crystals: effect ofthe position of the surface plane, cited above.

In contrast to surface modes, a core mode (e.g., a fundamental coremode) of an air-core PDF without a silica core ring is one in which thepeak of the mode intensity is located in the core. In most cases, mostof the energy will also be contained within the air core. Thepropagation constants of surface modes often fall close to or can evenbe equal to the propagation constant of the fundamental core mode. See,for example, K. Saitoh et al., Air-core photonic band-gap fibers: theimpact of surface modes, Douglas C. Allan et al., Surface modes and lossin air-core photonic band-gap fibers, in Photonic Crystals Materials andDevices, and Dirk Müller et al., Measurement of Photonic Band-gap FiberTransmission from 1.0 to 3.0 μm and Impact of Surface Mode Coupling,which are cited above.

The fundamental core mode generally couples quite strongly to one ormore of these surface modes by a resonant coupling mechanism or anearly-resonant coupling mechanism. Such coupling may be caused, forexample, by random (e.g., spatial) perturbations in the fiber indexprofile or cross-section. Since surface modes are inherently lossy dueto their high energy density in the dielectric of the fiber, suchcoupling is a source of propagation loss. Furthermore, since surfacemodes occur across the entire bandgap, no portion of the availablespectrum is immune to this loss mechanism. Recent findings havedemonstrated that surface modes are a cause of the reduced transmissionbandwidth in a 13-dB/km air-core PBF manufactured by Corning. See, forexample, N. Venkataraman et al., Low loss (13 dB/km) air core photonicband-gap fibre, Proceedings of European Conference on OpticalCommunication, ECOC 2002, Copenhagen, Denmark, PostDeadline Session 1,PostDeadline Paper PD1.1, Sep. 12, 2002; and C. M. Smith, et al.,Low-loss hollow-core silica/air photonic bandgap fibre, Nature, Vol.424, No. 6949, 7 Aug. 2003, pages 657-659, which are incorporated byreference herein. This effect is believed to be the source of theremaining loss (approximately 13 dB/km) in this air-corephotonic-bandgap fiber. See, for example, Douglas C. Allan et al,Photonic Crystals Materials and Devices, cited above. Understanding thephysical origin of surface modes and identifying fiber configurationsthat are free of such modes across the entire bandgap is therefore ofimportance in the ongoing search for low-loss PBFs.

SUMMARY OF THE INVENTION

In certain embodiments, an optical fiber comprises a cladding comprisinga first material having a first refractive index and a pattern of asecond material formed therein. The second material has a secondrefractive index lower than the first refractive index. The cladding hasa plurality of first regions that support intensity lobes of the highestfrequency bulk mode and has a plurality of second regions that do notsupport intensity lobes of the highest frequency bulk mode. The opticalfiber further comprises a central core region formed in the cladding.The optical fiber further comprises a core ring having an outerperimeter. The core ring surrounds the central core region, wherein theouter perimeter of the core ring passes only through the second regionsof the cladding.

In certain embodiments, an optical fiber comprises a cladding comprisinga dielectric material having a first refractive index and a periodicpattern of regions formed therein. Each region has a substantiallycircular cross-section and has a second refractive index lower than thefirst refractive index. Each region is spaced apart from adjacentregions. Each group of three regions adjacent to one another defines aportion of the dielectric material having a cross-section sized toenclose an inscribed circle having a circumference tangential to thethree adjacent regions. The optical fiber further comprises a coreregion formed in the cladding. The optical fiber further comprises acore ring having an outer perimeter. The core ring surrounds the coreregion, wherein the outer perimeter of the core ring does not passthrough any of the inscribed circles.

In certain embodiments, an optical fiber comprises a cladding comprisinga dielectric material having a first refractive index and having aperiodic pattern of regions formed therein. Each region has asubstantially circular cross-section. Each region has a secondrefractive index lower than the first refractive index. The opticalfiber further comprises a central core region formed in the cladding.The optical fiber further comprises a core ring having a generallycircular cross-section and an outer radius. The core ring surrounds thecentral core region, wherein the core ring induces ring surface modeshaving dispersion curves substantially decoupled from a fundamental modedispersion curve of the optical fiber.

In certain embodiments, an optical fiber comprises a cladding comprisinga dielectric material having a first refractive index and having aperiodic pattern of regions formed therein. Each region has asubstantially circular cross-section and has a second refractive indexlower than the first refractive index. The optical fiber furthercomprises a central core region formed in the cladding. The opticalfiber further comprises a core ring having a generally circularcross-section and a thickness. The core ring surrounds the central coreregion, wherein the thickness of the core ring is sufficiently small tosupport at most one ring-induced surface mode.

In certain embodiments, a method of designing an optical fiber isprovided. The optical fiber comprises a material with a pattern ofregions formed therein to form a cladding surrounding a core region. Thematerial has a first refractive index and the pattern of regions has asecond refractive index lower than the first refractive index. Themethod comprises designing a substantially circular core ring in thecladding. The core ring surrounds the core region and has an outerradius, an inner radius, and a thickness between the outer radius andthe inner radius. At least one of the outer radius, the inner radius,and the thickness is selected to reduce losses of the optical fiber.

In certain embodiments, a method of designing an optical fiber isprovided. The optical fiber comprises a material with a pattern ofregions formed therein to form a cladding surrounding a core region. Thematerial has a first refractive index and the pattern of regions has asecond refractive index lower than the first refractive index. Themethod comprises designing a substantially circular core ring in thecladding. The core ring surrounds the core region and has an outerradius, an inner radius, and a thickness between the outer radius andthe inner radius. At least one of the outer radius, the inner radius,and the thickness is selected to reduce the number of ring-inducedsurface modes.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a partial cross section of an exemplarytriangular-pattern air-core photonic-bandgap fiber (PBF) for a coreradius of 1.15Λ and a hole radius ρ of approximately 0.47Λ.

FIG. 2 illustrates an enlarged view of the partial cross section of FIG.1 to provide additional detail on the spatial relationships between theair holes, the segments (membranes) between adjacent air holes and theveins (corners) at the intersections of the segments.

FIG. 3 illustrates contour lines that represent equal intensity lines ofa typical surface mode for the air-core PBF of FIG. 1.

FIG. 4 illustrates contour lines that represent equal intensity lines ofthe fundamental core mode for the air-core PBF of FIG. 1.

FIG. 5 illustrates contour lines that represent equal intensity lines ofa typical bulk mode for the triangular-pattern air-core PBF of FIG. 1,but without the removal of the central structure to form the air core106.

FIG. 6 illustrates dispersion curves of the defect modes for theair-core photonic-bandgap fiber (PBF) of FIG. 1 having atriangular-pattern of holes with a photonic-crystal structure of period(i.e., hole-to-hole spacing) A and a hole radius ρ of approximately0.47Λ, surrounding an air-core having a radius R of approximately 1.15Λ,wherein the shaded (cross hatched) area represents the photonic bandgapof the crystal.

FIG. 7 illustrates dispersion curves of the defect modes for an air-corePBF having a core radius R of approximately 1.8Λ.

FIG. 8 illustrates a partial cross section showing the hole pattern andair-core shape of a PBF from which the dispersion curves of FIG. 7 areobtained.

FIG. 9 illustrates a graph of the number of core modes (diamonds) andsurface modes (triangles) versus the air-core radius at the normalizedfrequency ωΛ/2πc=1.7.

FIGS. 10A, 10B and 10C illustrate the core shapes for core radii of0.9Λ, 1.2Λ, and 2.1Λ, respectively, from which the information in FIG. 9was derived.

FIG. 11 illustrates a graphical representation of the air-core radiusranges that support core modes only (unshaded rings) and both core andsurface modes (shaded rings).

FIG. 12 illustrates the partial cross section of the triangular-patternair-core PBF of FIG. 1 with a core of radius R₁ formed in the photoniccrystal lattice, wherein the surface of the core intersects the cornersof the photonic crystal lattice and wherein surface modes are supported.

FIG. 13 illustrates the partial cross section of the triangular-patternair-core PBF of FIG. 1 with a core of radius R₂ formed in the photoniccrystal lattice, wherein the surface of the core does not intersect thecorners of the photonic crystal lattice and wherein surface modes arenot supported.

FIG. 14 illustrates a plot (dotted curve) of the maximum intensity ofthe highest frequency bulk mode on a circle of radius R as a function ofR overlaid on the plot (solid curve) of the maximum number of surfacemodes as a function of R from FIG. 9.

FIGS. 15A and 15B illustrate intensity contour maps of the two highestfrequency doubly degenerate bulk modes below the bandgap at the Γ point,wherein R₁ is an example of a core radius that supports both core modesand surface modes, and R₁ is an example of a core radius that supportsonly core modes.

FIG. 16 illustrates a graphical representation of a partial crosssection of the triangular pattern air core PBF, wherein black circles ateach dielectric corner represent dielectric rods, and wherein unshadedrings represent bands of core radii for which the surface of the coredoes not intersect the dielectric rods.

FIG. 17 illustrates a graph (dashed lines) of the results of thenumerical simulations of the number of surface modes and illustrates agraph (solid lines) of the number of surface modes predicted using thegeometric model of FIG. 16 and counting the number of rods intersectedby the surface of the core, wherein the number of surface modes in eachgraph is plotted with respect to the normalized core radius R/Λ.

FIG. 18 illustrates a plot of the normalized core radius R/Λ versus thenormalized hole radius ρ/Λ to show the effect of the fiber air-fillingratio on the presence of surface modes.

FIG. 19 schematically illustrates a cross section of an alternativeair-core photonic bandgap fiber having a non-circular (e.g., hexagonal)core shape and no surface modes.

FIGS. 20A and 20B illustrate for comparison the effective refractiveindices of core modes and surface modes for two commercially availablephotonic bandgap fibers.

FIG. 21A illustrates a cross-section of an exemplary air-corephotonic-bandgap fiber with a central core of radius R=0.9Λ, such thatthe core does not support surface modes.

FIG. 21B illustrates a cross-section of an exemplary fiber similar tothe fiber of FIG. 21A also having a central core of radius R=0.9Λ, butwith a thin silica ring around the core.

FIG. 22 illustrates a cross-section of a generic preform for an air-corephotonic-bandgap fiber, wherein the preform comprises a stack of silicatubes with seven center tubes removed to form the single-mode core ofthe fiber.

FIG. 23A illustrates a calculated ω-k diagram of the air-core fiber ofFIG. 21A, which does not have the thin silica ring around the centralcore.

FIG. 23B illustrates a calculated ω-k diagram of the air-core fiber ofFIG. 21B, which has the thin silica ring around the central core.

FIG. 24A illustrates intensity contour lines of the fundamental coremode of the air-core fiber of FIG. 21A (no ring around the core)calculated at k_(z)Λ/2π=1.7, wherein the relative intensity on thecontours varies from 0.1 to 0.9 in increments of 0.1.

FIG. 24B illustrates intensity contour lines of the fundamental coremode of the air-core fiber of FIG. 21B (ring around the core) alsocalculated at k_(z)Λ/2π=1.7, wherein the relative intensity on thecontours varies from 0.1 to 0.9 in increments of 0.1.

FIGS. 25A and 25B illustrates the intensity contour lines of twoexemplary surface modes of the fiber of FIG. 21B to illustrate theeffect of the thin silica ring around the core.

FIG. 26A illustrates a cross-section of an air-core photonic-bandgapfiber with a central air-core with no ring around the core, the air-corehaving a radius R=1.13Λ, such that the core supports surface modes.

FIG. 26B illustrates a cross-section of an air-core photonic-bandgapfiber with a central air-core with a thin silica ring around the core,the air-core also having a radius R=1.13Λ.

FIG. 27A illustrates a calculated ω-k diagram of the air-core fiber (noring around the core) of FIG. 26A.

FIG. 27B illustrates a calculated ω-k diagram of the air-core fiberhaving a thin silica ring around the core of FIG. 26B.

FIG. 28 illustrates a cross section of a photonic crystal terminated bya thin slab for modeling a very thin PBF core ring.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

Certain embodiments disclosed herein are based on information obtainedin an investigation of the properties of the core modes and the surfacemodes of PBFs using numerical simulations. The investigation focused onthe most common PBF geometry, namely fibers with a periodic, triangularpattern of cylindrical air-holes in the cladding and a circular coreobtained by introducing an air defect. Such fibers are described, forexample, in R. F. Cregan et al., Single-Mode Photonic Band Gap Guidanceof Light in Air, cited above; Jes Broeng et al., Analysis of air-guidingphotonic bandgap fibers, cited above; and Jes Broeng et al., PhotonicCrystal Fibers. A new class of optical waveguides, Optical FiberTechnology, cited above. The results are also applicable to a broadrange of air-hole patterns (e.g., hexagonal patterns, square patterns,etc.), hole shapes, core shapes, and core ring shapes. The results arealso applicable to other photonic-bandgap fibers, namely, fiber withsimilar geometries that operate on the same photonic-bandgap principlebut with a core not necessarily filled with air (e.g., a core filledwith another gas, a vacuum, a liquid, or a solid), with cladding holesnot necessarily filled with air (e.g., cladding holes filled withanother gas, a vacuum, a liquid, or a solid), and with solid portions ofthe cladding or the core ring not necessarily made of silica (e.g.,another solid or a multiplicity of solids). As used herein, hole or acore that is not filled with a solid or a liquid is referred to hereinas being hollow. It is understood here that the respective refractiveindices of the materials that make up the core, the cladding holes, andthe solid portion of the cladding are selected in certain embodimentssuch that the fiber structure supports a guided mode via thephotonic-bandgap effect. This implies that the refractive index of thecore and the refractive index of the holes is lower than that of therefractive index of the solid portions of the cladding, and that thedifference between these indices is large enough to support the guidedmode.

New geometries are proposed herein for air-core fibers or fibers with acore that has a lower refractive index than the solid portions of thecladding. In certain embodiments, these geometries have ranges of corecharacteristic dimensions (e.g., core radii when the core is circular)for which the fiber core has reduced propagation losses due to surfacemodes. In particular, for certain embodiments having a circular corewith a radius between about 0.7Λ and about 1.05Λ, where A is thehole-to-hole spacing of the triangular pattern, the core supports asingle mode and does not support any surface modes. The absence orreduction of surface modes suggests that fibers in accordance withcertain embodiments described herein exhibit substantially lower lossesthan current fibers. As further shown below, the existence of surfacemodes in the defect structure can be readily predicted either from astudy of the bulk modes alone or even more simply by a straightforwardgeometric argument. Because the structure is truly periodic, predictionof the existence of surface modes in accordance with the methodsdescribed below is quicker and less complicated than a full analysis ofthe defect modes.

Photonic-Bandgap Fibers with No Core Ring

In certain embodiments, the methods disclosed herein can be used topredict whether a particular fiber geometry will support surface modesso that fibers can be designed and manufactured that do not supportsurface modes or that support only a reduced number of surface modes. Inparticular, in certain embodiments, the presence of surface modes can beavoided or reduced by selecting the core radius or other characteristicdimension such that the edge of the core does not cut through any of thecircles inscribed in the veins (e.g., the solid intersection regions) ofthe PBF lattice. The technique works for broad ranges of geometries andhole sizes.

In order to avoid or reduce surface modes, certain embodiments of thetechniques described herein are used to design the core shape such thatthe core does not intersect any of the veins of the PBF lattice (e.g.,the core intersects only the segments that join the veins of the PBFlattice). By following this general criterion, PBFs can be designed tobe free of surface modes.

Certain embodiments described herein are based on a photonic band-gapfiber (PBF) with a cladding photonic crystal region comprising atriangular lattice comprising a plurality of circular holes filled witha gas (e.g., air) in silica or other solids, where the holes are spacedapart by a period A. See, e.g., R. F. Cregan et al., Single-ModePhotonic Band Gap Guidance of Light in Air, cited above; Jes Broeng etal., Analysis of air-guiding photonic bandgap fibers, cited above; andJes Broeng et al., Photonic Crystal Fibers. A New Class of OpticalWaveguides, cited above. For simplicity, such fibers are referred toherein as air-hole fibers; however, as discussed above, the followingdiscussions and results are also applicable to photonic-bandgap fiberswith a core and/or all or some of the cladding holes filled with othermaterials besides air (e.g., another gas, a vacuum, a liquid, or asolid) and with solid portions of the cladding made of materials otherthan silica (e.g., a different solid or a multiplicity of solids).Furthermore, the results are also adaptable to other patterns of holes(e.g., hexagonal patterns, square patterns, etc.).

A partial cross section of an exemplary triangular-pattern air-core PBF100 is illustrated in FIG. 1. As illustrated, the fiber 100 comprises asolid dielectric lattice 102 comprising a plurality of air holes 104surrounding an air core 106. Three exemplary adjacent holes 104 areshown in more detail in FIG. 2. The portion of the solid lattice 102between any three adjacent holes 104 is referred to as a vein (or acorner) 110, and the thinner regions connecting two adjacent veins(i.e., a region between any two adjacent holes) is referred to as asegment (or a membrane) 112. In the illustrated embodiment, each airhole 104 has a radius ρ. The center-to-center spacing of adjacent airholes 104 is referred to as the period Λ of the photonic crystal.

As will be discussed in more detail below, each vein 110 can beapproximated by an inscribed circle 114 of radius α, wherein thecircumference of the inscribed circle 114 is tangential to thecircumferences of three holes 104 surrounding the vein 110. Simplegeometric calculations readily show that the radius a of the inscribedcircle 114 is related to the radius ρ and the period Λ of the air holes104 as follows:

$a = {\left( {\Lambda/\sqrt{3}} \right) - \rho}$

As illustrated in FIG. 1, the air-core 106 of the PBF 100 isadvantageously created by introducing a larger cylindrical air hole ofradius R at the center of the fiber. The location of this cylinder,reproduced in FIG. 1 as a dashed circle, is referred to herein as theedge of the core 106. The radius R is referred to herein as thecharacteristic dimension of the air-core 106. In the example of thecircular core illustrated in FIG. 1, the radius R is the radius of thecircular core The following discussion is adaptable to cores havingother shapes and characteristic dimensions (e.g., the shortest distancefrom the center to the nearest boundary of a polygonal-shaped core). Inthe PBF 100 of FIGS. 1 and 2, the radius R is selected to be 1.15Λ, andthe radius ρ of each air hole 104 is selected to be 0.47Λ. For example,the air-core 106 of radius 1.15Λ is advantageously selected because thecore radius corresponds to a core formed in practice by removing sevencylinders from the center of the PBF preform (e.g., effectively removingthe glass structure between the seven cylinders). Such a configurationis described, for example, in J. A. West et al., Photonic CrystalFibers, Proceedings of 27^(th) European Conference on OpticalCommunications (ECOC'01-Amsterdam), Amsterdam, The Netherlands, Sep.30-Oct. 4, 2001 paper ThA2.2, pages 582-585, which is herebyincorporated herein by reference. Other values of the radius R (e.g.,between approximately 0.7Λ and approximately 1.2Λ) and the radius ρ(e.g., between approximately 0.49Λ and 0.5Λ) are also compatible withembodiments described herein.

As discussed above, surface modes are defect modes that form at theboundary between the core 106 and the photonic-crystal cladding 102. Atypical surface mode for the triangular-pattern air-core PBF 100 ofFIGS. 1 and 2 is illustrated in FIG. 3. A typical fundamental core modefor the PBF 100 of FIGS. 1 and 2 is illustrated in FIG. 4. In FIGS. 3and 4, the contour lines represent equal intensity lines. The outmostintensity line in each group has a normalized intensity of 0.1 and theinnermost intensity line has a normalized intensity of 0.9, and eachintervening intensity line represents a normalized step increase of 0.1.

In the absence of a core, a PBF carries only bulk modes. An example ofbulk mode is illustrated in FIG. 5. The bulk mode of FIG. 5 iscalculated for the same triangular-pattern air-core PBF 100 illustratedin FIG. 1, but without the removal of the central structure to form theair core 106. As in FIGS. 3 and 4, the contour lines in FIG. 5 representequal intensity lines.

The particular bulk mode illustrated in FIG. 5 comprises a series ofnarrow intensity lobes centered on each of the thicker dielectriccorners 110 of the photonic crystal 102. Other bulk modes may havedifferent lobe distributions (e.g., all the lobes may be centered onmembranes 112 rather than on corners 110).

As discussed above, a fiber will support many surface modes unless thefiber is suitably designed to eliminate or reduce the number of surfacemodes. As further discussed above, the propagation constants of thesurface modes are often close to or equal to the propagation constant ofthe fundamental core mode, and, as a result, the core mode can easily becoupled to the surface modes (e.g., by random perturbations in the fibercross section), which results in an increased propagation loss for thefundamental core mode. This problem is also present for other core modesbesides the fundamental mode when the fiber is not single mode.

By varying the radius R of the air core 106, the effect of the coreradius on the core modes and the effect of surface truncation on thesurface mode behavior can be systematically studied. One such study isbased on simulations performed on the University of Michigan AMD Linuxcluster of parallel Athion 2000MP processors using a full-vectorialplane-wave expansion method. An exemplary full-vectorial plane waveexpansion method is described, for example, in Steven G. Johnson et al.,Block-iterative frequency-domain methods for Maxwell's equations in aplanewave basis, Optic Express, Vol. 8, No. 3, 29 Jan. 2001, pages173-190, which is hereby incorporated herein by reference.

The simulations disclosed herein used a grid resolution of Λ/16 and asupercell size of 8Λ×8Λ. The solid portion of the cladding was assumedto be silica, and all holes were assumed to be circular and filled withair. When running the simulations with 16 parallel processors, completemodeling of the electric-field distributions and dispersion curves ofall the core modes and surface modes of a given fiber typically takesbetween 7 hours and 10 hours.

The results of the simulation for a triangular pattern indicate that aphotonic bandgap suitable for air guiding exists only for air-hole radiiρ larger than about 0.43Λ. In certain embodiments, the largest circularair-hole radius that can be fabricated in practice (e.g., so thatsufficient silica remains in the membranes 112 between adjacent airholes 104 to provide a supporting structure) is slightly higher than0.49Λ. In certain embodiments described herein, a structure is simulatedthat has an air-hole radius ρ between these two extreme values. Inparticular, ρ is selected to be approximately 0.5Λ. Although thesimulations described herein are carried out for ρ=0.47Λ, similarresults have been obtained for any value of ρ between 0.43Λ to 0.5Λ, andthe qualitative conclusions described herein are valid for any air-holesize in the range of 0.43Λ to 0.5 Λ.

FIG. 6 illustrates the theoretical ω-k_(z) diagram of the fiber geometryunder study generated for a core radius R=1.15Λ (see, for example, FIG.1). In FIG. 6, the vertical axis is the optical angular frequencyω=2πc/λ normalized to 2πc/Λ (i.e., Λ/λ), where λ is the free-spacewavelength of the light signal, c is the velocity of light in vacuum,and Λ is the photonic-crystal structure period. Thus, the vertical axisrepresents ωΛ/2πc=Λ/λ, which is a dimensionless quantity. The horizontalaxis in FIG. 6 is the propagation constant along the axis of the fiber(z direction) k_(z), normalized to 2π/Λ (i.e., k_(z)Λ/2π).

The first photonic bandgap supported by the infinite structure of thesimulated fiber 100 of FIG. 1 is represented by the shaded(cross-hatched) region. The size and shape of the first photonic bandgapdepends on the value of the radii ρ of the air holes 104 (which areequal to 0.47Λ in the illustrated simulation), but the bandgap is verynearly independent of the dimension of the core 106. The dashed line inFIG. 6 represents the light line, below which no core modes can exist,irrespective of the core size and the core shape. The portion of theshaded region above the dashed line shows that in the simulated fiber100, the normalized frequencies for which light can be guided in the aircore range from approximately 1.53 to approximately 1.9.

The solid curves in FIG. 6 represent the dispersion relations of thecore mode and the surface modes. The air core actually carries twofundamental modes. Each mode is nearly linearly polarized, and thepolarization of each mode is orthogonal to the polarization of the othermode. These two modes are very nearly degenerate. In other words, thetwo modes have almost exactly the same dispersion curve within thebandgap. The topmost curve in FIG. 6 actually comprises two dispersioncurves, one for each of these two fundamental modes; however, the twocurves are so nearly identical that they cannot be distinguished on thisgraph. The related intensity profiles of selected modes at k_(z)Λ/2π=1.7are plotted in FIG. 4 for one of the two fundamental core modes and inFIG. 3 for an exemplary surface mode. These profiles indicate that thehighest-frequency modes inside the bandgap are the two fundamental coremodes. All other modes in the bandgap are surface modes, which havetheir intensities localized at the core-cladding boundary, as shown inFIG. 3. The strength of the spatial overlap with the silica portions ofthe fiber is different for core and surface modes. The difference instrength results in the core mode having a group velocity close to c andthe surface modes having a lower group velocity, as illustrated in FIG.6.

FIG. 6 also illustrates another distinguishing feature of the core andsurface modes. In particular, the curves for the surface modes alwayscross the light line within the bandgap. In contrast, the curves for thecore modes never cross the light line within the bandgap.

The behaviors of the core mode and the surface modes are investigated asa function of defect size by changing the core radius R from 0.6Λ to2.2Λ in 0.1Λ steps. FIG. 7 illustrates the ω-k_(z) diagram generated forthe same fiber geometry as used to generate the information in FIG. 6,but for a larger core radius (R=1.8Λ). As illustrated by the partialfiber cross section in FIG. 8, the larger core radius is formed, forexample, by removing additional lattice structure beyond the centralseven cylinders of the preform so that the surface of the core 106intersects the thinner membranes 112 between the holes 104 rather thanintersecting the thicker dielectric corners 110. As expected, the numberof core modes appearing in FIG. 7 for the embodiment of FIG. 8 isgreater than for the embodiment of FIG. 1. In addition, all the modesare core modes for this larger radius. As the frequency is increasedfrom the low-frequency cutoff of the bandgap, the highest order coremodes appear first, in a group of four or more modes (e.g., four in FIG.7). This feature depends on the core size and mode degeneracy. See, forexample Jes Broeng et al., Analysis of air-guiding photonic bandgapfibers, cited above. As the frequency is further increased, the numberof modes reaches some maximum number (14 in the example illustrated inFIG. 7) at a normalized frequency (ωΛ/2πc) of approximately 1.7. Above anormalized frequency of approximately 1.7, the number of modes graduallydecreases to two (the two fundamental modes) at the high-frequencycutoff of the bandgap. The maximum number of core modes occurs at or inthe vicinity of the frequency where the light line intersects the lowerband edge. In the embodiment illustrated by the plot in FIG. 7, thelight line intersects the lower band edge at a normalized frequency(ωΛ/2πc) having a value of around 1.67. Note that in FIG. 7, many of thecurves represent multiple modes that are degenerate and thus overlap inthe diagram.

FIG. 9 illustrates the dependence of this maximum number of core modes(i.e., the number of modes is plotted at ωΛ/2πc=1.7) on R. The number ofsurface modes is also shown in FIG. 9. In addition, core shapes forrepresentative radii of R=0.9Λ, R=1.2Λ, and R=2.1Λ are illustrated inFIG. 10A, FIG. 10B and FIG. 10C, respectively. As stated above, the gridresolution used to generate the data points in FIG. 9 was Λ/16. However,to generate additional points in the more interesting range of coreradii between 1.1Λ and 1.3Λ, the grid size was reduced to Λ/32 in thatrange. As a result, the absolute number of surface modes predicted inthis range does not scale the same way as in the rest of the graph. Thisis inconsequential since the primary interest in generating the datapoints is to determine the boundaries of the surface mode regions.

The behaviors of the core modes in PBFs and in conventional fibers basedon total internal reflection have striking similarities. The fundamentalmode, like an LP₀₁ mode, is doubly degenerate (see FIGS. 6 and 7), isvery nearly linearly polarized, and exhibits a Gaussian-like intensityprofile. See, for example, Jes Broeng et al., Analysis of air-guidingphotonic bandgap fibers, cited above. The next four modes are alsodegenerate, and the electric field distributions of these four modes arevery similar to those of the HE₂₁ ^(odd), HE₂₁ ^(even), TE₀₁, and TM₀₁modes of conventional fibers. Many of the core modes, especially thelow-order modes, exhibit a two-fold degeneracy in polarization over muchof the bandgap. As the core radius is increased, the number of coremodes increases in discrete steps (see FIG. 9), from two (the twofundamental modes) to six (these two modes plus the four degeneratemodes mentioned above), then 14 (because the next eight modes happen toreach cutoff at almost the same radius), etc.

FIG. 9 also illustrates another aspect of the modes. In particular, whenR falls in certain bounded ranges, all modes are found to be core modes.The first three of the bounded ranges are:

range 1 from approximately 0.7Λ to approximately 1.1Λ;

range 2 from approximately 1.3Λ to approximately 1.4Λ; and

range 3 from approximately 1.7Λ to approximately 2.0Λ.

FIG. 7 illustrates the case where R is equal to 1.8Λ, which is oneparticular example of a surface-mode-free PBF in range 3. Thesurface-mode-free ranges determined by the computer simulation areillustrated schematically in FIG. 11. In FIG. 11, the background patternof circles represents the infinite photonic crystal structure, the fourshaded (cross hatched) annular areas represent the ranges of core radiithat support surface modes, and the three unshaded annular areas(labeled as band 1, band 2 and band 3) represent the first three rangesof radii that are free of surface modes. Note that for radii less than0.5Λ (e.g., the central unshaded portion of FIG. 11), the core does notsupport core modes that are guided by the photonic-bandgap effect.

FIG. 11 is simply a different way to graph the regions of no surfacemodes shown in FIG. 9. Thus, the three ranges of radii in FIG. 9 thatsupport no surface modes, as shown by the open triangles that fall alongthe bottom horizontal axis, are graphed as the three white annular(unshaded) regions in FIG. 11 (bands 1, 2 and 3). The complementary(shaded) bands between the white bands correspond to the ranges of radiiin FIG. 9 where the triangles are above the horizontal axis and thusrepresent radii that support surface modes.

In the first of the unshaded ranges in FIG. 11 (e.g., band 1 fromapproximately 0.7Λ to approximately 1.1Λ), the core supports a singlecore mode and does not support any surface modes at all across theentire wavelength range of the bandgap, i.e., the PBF is truly singlemode. There do not appear to be any previous reports of a single-modeall-silica PBF design in the literature. Note that in band 2, band 3 andall other bands representing larger radii, the fiber is no longer singlemode.

An example of a terminating surface shape that falls in this single-moderange (e.g., range 1) is shown in FIG. 10A for R equal to 0.9Λ. Theseparticular configurations may be fabricated using small tips of glassprotruding into the core using an extrusion method and other knownfabrication techniques.

The number of surface modes is also strongly dependent on the coreradius, albeit in a highly non-monotonic fashion. For core radii in thevicinities of approximately 0.6Λ, approximately 1.2Λ, approximately1.6Λ, and approximately 2.1Λ, many surface modes are introduced,resulting in the peaks in the number of surface modes. The peaks areapparent in FIG. 9. Moreover, in these vicinities, the number of surfacemodes varies rapidly with R. Typical experimental PBFs are fabricated byremoving the central 7 cylinders (R approximately equal to 1.15Λ) or 19cylinders (R approximately equal to 2.1Λ) from the preform to form thecore 106; however, these particular values of R, which happen to be morestraightforward to manufacture, also happen to lead to geometries thatsupport surface modes, as shown, for example, in FIG. 9.

Based on the foregoing results of the computer simulations, the basicconditions at which surface modes occur have been investigated andcertain embodiments described herein have no surface modes. The basicconditions lead to the observation that surface modes are created whenthe surface of the core 106 intersects one or more of the dielectriccorners 110 of the photonic crystal lattice 102. From this observation,a fast and simple geometric criterion is obtained for evaluating whethera particular fiber configuration supports surface modes. As discussedbelow, when the geometric criterion is applied to triangular-patternPBFs 100 with a circular air core 106, the approximate geometric modelyields quantitative predictions in acceptable agreement with the resultsof computer simulations described above.

As discussed above, surface modes can occur when an infinite photoniccrystal is abruptly terminated, as happens for example at the edges of acrystal of finite dimensions. For example, in photonic crystals made ofdielectric rods in air, surface modes are induced only when thetermination cuts through rods. A termination that cuts only through airis too weak to induce surface modes.

In an air-core PBF 100, the core 106 also acts as a defect that perturbsthe photonic crystal lattice 102 and may introduce surface modes at theedge of the core 106. Whether surface modes appear, and how many appear,depends on how the photonic crystal is terminated, which determines themagnitude of the perturbation introduced by the defect. In the absenceof an air core, a PBF carries only bulk modes, as discussed above withrespect to FIG. 5.

When the air core 106 is introduced as shown in FIGS. 1, 3 and 4, thecore 106 locally replaces the dielectric material of the photoniccrystal lattice 102 with air. The portions of the surface of the core106 that cut through the cladding air holes 104 in FIG. 1 replace air byair. Thus, just as in the case of a planar photonic crystal (asdescribed, for example, in J. D. Joannopoulos et al., Photonic Crystals:Molding the flow of light, cited above), those portions of the coresurface do not induce significant perturbation. Only the portions of thecore surface that cut through the dielectric corners 110 or thedielectric membrane 112 of the photonic crystal lattice 102 in FIG. 1replace dielectric by air and thereby perturb the bulk modes of FIG. 5.Whether the perturbation is sufficient to potentially induce surfacemodes, such as the surface modes shown in FIG. 3, is discussed below.

Since a core 106 of any size and shape always cuts through somedielectric material, some perturbation is always introduced by the core106. The sign of the perturbation is such that in the ω-k diagram, thebulk modes are all shifted up in frequency from their frequencies intheir respective unperturbed positions. For a silica/air PBF 100, theperturbation is comparatively weak, and the frequency shift is smallsuch that almost all perturbed bulk modes remain in a bulk mode band.Exceptions to the foregoing are modes from the highest frequencybulk-mode band of the lower band (referred to hereinafter as “HFBM”).Because such modes are located just below the bandgap in the ω-kdiagram, the perturbation moves them into the bandgap as surface modes.See, for example, J. D. Joannopoulos et al., Photonic Crystals: Moldingthe flow of light, cited above.

Surface modes can be written as an expansion of bulk modes. For the weakperturbation considered here, it can be shown that the main term in thisexpansion is the HFBM, as expected in view of the origin of thesesurface modes. The HFBM is the bulk mode illustrated in FIG. 5. Asillustrated in FIG. 5, the lobes of the mode are all centered on corners110 of the crystal 102, which results in two important consequences.First, because surface modes are induced by a perturbation of this bulkmode, the lobes of the surface modes are also centered on the corners110, as shown, for example, in FIG. 3. Second, for the HFBM to beperturbed and yield surface modes, the perturbation must occur indielectric regions of the photonic crystal lattice 102 that carry asizable HFBM intensity, e.g., in regions at the corners 110 of thephotonic crystal 102. These observations show that surface modes arestrongly correlated with the magnitude of the perturbation introduced bythe air core 106 on the HFBM. If the surface of the core 106 intersectslobes of the HFBM at the corners 110 of the dielectric lattice 102 (asillustrated, for example, by a core of radius R₁ in FIG. 12), theperturbation is large and surface modes are induced. The number ofsurface modes then scales like the highest intensity intersected by thecore 106 in the dielectric 102. Conversely, if the surface of the core106 does not intersect any of the lobes of this bulk mode (asillustrated, for example, by a core of radius R₂ in FIG. 13), no surfacemodes are created.

The foregoing is illustrated in FIG. 14, which reproduces the plot ofthe number (values on the left vertical axis) of surface modes atωΛ/2πc=1.7 on a circle of radius R as a function of R normalized to Λ(horizontal axis) as a solid curve. FIG. 14 also includes a plot (dottedcurve) of the maximum intensity (values in arbitrary units on the rightvertical axis) of the highest frequency bulk mode. FIG. 14 clearly showsthe relationship between the maximum intensity and the number of surfacemodes. The two curves in FIG. 14 are clearly strongly correlated, whichconfirms that surface modes occur for radii R such that the edge of thecore cuts through high-intensity lobes of the highest frequency bulkmode. Based on this principle, a first approximate dependence of thenumber of surface modes on the core radius was developed. By comparisonto the results of exact simulations, the foregoing shows that theresults obtained using this HFBM criterion predicts the presence orabsence of surface modes fairly accurately. Of course, many other kindsof perturbations can induce surface modes in the photonic crystal 102,so that the foregoing condition for the absence of surface modes is anecessary condition but it is not always a sufficient condition.

In one criterion for determining the presence of the surface modes, theelectromagnetic intensity of the highest frequency bulk modes isintegrated along the edge of the core. It is sufficient to perform suchintegration for either one of the two doubly degenerate modes, since theintegrations for both modes are equal, as required by symmetry.

The foregoing determination of the radius R of the air core can beperformed in accordance with a method of numerically computing theintensity distribution of the bulk modes of the infinite fiber cladding.In accordance with the method, the intensity distribution of the highestfrequency bulk mode of the fiber of interest without the air core isfirst determined. Thereafter, a circular air core of radius R issuperposed on that intensity distribution. As illustrated in FIGS. 15Aand 15B, changing the core radius R causes the edge of the core to passthrough different areas of this field distribution. In accordance withthe computing method, the fiber will support surface modes when the edgeof the core intersects high lobe regions of this field distribution. InFIGS. 15A and 15B, a core of radius R=R₁ is one example of a core radiusthat passes through several (six in this example) high intensity lobesof the highest frequency bulk mode. The computing method predicts that acore with such a radius will support surface modes. At the otherextreme, when the core has a radius R=R₂, as illustrated in FIGS. 15Aand 15B, the core edge does not pass through any of the high-intensitylobes of the bulk mode, and such a core of radius R₂ does not supportsurface modes.

Although, described in connection with a circular core, it should beunderstood that the foregoing method is not limited to circular cores,and the method is applicable to any core shape.

As described above, the computing method is qualitative. In accordancewith the method, if the edge of a core of a selected radius R intersectshigh intensity lobes of the bulk mode, the fiber having a core of thatradius will support surface modes. As described thus far, the methoddoes not stipulate how many surface modes are supported. Furthermore,the method does not specify how high an intensity must be intersected bythe edge of the core or how many high intensity lobes the edge of thecore must intersect before surface modes appear (i.e., are supported).

The HFBM criterion is advantageously simplified by recognizing that theintensity lobes of the HFBM are nearly azimuthally symmetric, as shownin FIG. 5. Thus, the portion of each lobe confined in a dielectriccorner 110 can be approximated by the circle 114 inscribed in the corner110, as illustrated in FIG. 2. As discussed above, the radius a of theinscribed circle 114 is related to the period Λ and radius ρ of theholes 104 of the triangular pattern by α=(Λ/√3)−ρ.

The portions of the HFBM confined to the dielectric are approximated bya two-dimensional array of circles 114 centered on all thephotonic-crystal corners 110, as illustrated in FIG. 16, which isplotted for a triangular pattern and ρ=0.47Λ. This approximation enablesa new, simpler existence criterion to be formulated for surface modes:surface modes exist when and only when the surface of the core 106intersects one or more of the circles 114. Of course, many other kindsof perturbations can induce surface modes in the photonic crystal 102,so that the foregoing condition for the absence of surface modes is anecessary condition but it is not always a sufficient condition.

The same geometric criterion can also be derived using coupled-modetheory. In view of the symmetry of the lower-band bulk modes, eachcorner 110 can be approximated by a dielectric rod inscribed in thecorner 110, wherein the rod extends the length of the PBF 100. Eachisolated rod is surrounded by air and constitutes a dielectricwaveguide. The dielectric waveguide carries a fundamental mode withstrong fields in the rod that decay evanescently into the surroundingair, so the field looks much like the individual lobes of the HFBMillustrated in FIG. 5. Thus, the periodic array of rods has the patternof the circles 114 illustrated in FIG. 16. The waveguide modes of theindividual rods are weakly coupled to each other due to the proximity ofneighboring rods and form the bulk modes.

The HFBM is just one particular superposition of individual waveguidemodes. If an air core 106 that cuts into one or more rods is introduced,the removal of dielectric perturbs the waveguide modes in the oppositedirection to that forming bulk modes. The waveguide modes of the ring ofperturbed rods intersected by the surface of the core 106 are thencoupled to each other and form a surface mode. This surface mode issupported by the ring of rods and has fields that decrease outside eachrod, as evidenced by the exemplary surface mode of FIG. 3. If thesurface of the core 106 cuts only through membranes 112 instead ofcorners 110, the rods are unperturbed, and the modes couple to eachother much as they did without the presence of the core 106. Thus, nosurface mode is formed. In accordance with this description, surfacemodes exist if and only if the surface of the core 106 intersects rods.This is the same criterion that was derived above by approximating theHFBM lobes by the inscribed circles 114.

To verify the validity of this new geometric criterion, the criterion isapplied to the most widely studied class of air-core PBFs, namely fiberswith circular air holes in a triangular pattern, as illustrated in FIG.16. The core 106 is a larger circular air hole of radius R at the centerof the fiber 100. Again, this analysis postulates that when R isselected so that the surface of the core 106 intersects one or more rods(e.g., the circles 114 in FIG. 16), then surface modes will exist, andthe number of surface modes will be proportional to the number of rodsintersected. This scaling law is expected because as the number ofintersected rods increases the perturbation magnitude increases and thenumber of surface modes also increases. Conversely, when the surface ofthe core 106 does not intersect any rods, no surface modes occur. Asimple diagram of the fiber cross section, such as the diagramillustrated in FIG. 16, makes the application of this criterion to anyfiber geometry very easy.

The result of the foregoing geometric analysis is graphed in FIG. 16 fora triangular pattern. The shaded (cross hatched) rings in FIG. 16represent the ranges of core radii that intersect rods and thus supportsurface modes. As discussed above with respect to FIG. 11, the unshadedrings between the shaded rings (band 1-band 6) represent ranges of radiithat intersect no rods and thus do not support surface modes. Thedependence of the number of surface modes on the core radius iscalculated straightforwardly by applying elementary trigonometry to FIG.16 to determine the number of rods crossed by the surface of a core 106of a given radius. The numbers are plotted as a solid curve in FIG. 17,wherein the horizontal axis of the graph is the core radius normalizedto the crystal period Λ (e.g., R/Λ), and wherein the left vertical axisrepresents the number of rods intersected by the surface of the core, aspredicted by the geometric criterion.

The simple postulate predicts the important result illustrated in FIG.17 that several bands of radii for this type of PBF 100 support nosurface modes at all across the entire bandgap. Six such bands occur inthe range covered in FIG. 17 for radii R up to 3.5Λ, where Λ is thecrystal period as defined above. The range in FIG. 17 does not encompassthe band below R=0.47Λ, for which the radii are too small to support acore mode. Although not shown in FIG. 17, another eight bands occur forradii larger than 3.5Λ. The last band is at R approximately equal to8.86Λ.

Table 1 lists the boundaries and the widths of 14 bands of core radiithat support no surface modes in triangular PBFs with ρ=0.47Λ. As shownin Table 1, the first band is the widest. The first band is also themost important for most purposes because the first band is the only bandthat falls in the single-mode range of this PBF 100 (e.g., in the rangewhere R is less than about 1.2 for an air-hole radius ρ equal to 0.47Λ).All other bands, except for the third one, are substantially narrower.Generally, the bands where no surface modes are supported becomenarrower as the radius of the core 106 increases. Note that by nature ofthe rod approximation, these values are independent of the refractiveindex of the photonic crystal lattice dielectric 102.

TABLE 1 Range from Range from geometric HFBM Width criterion criterionof Band Band (in units (in units Range from simulations (in units No. ofΛ) of Λ) (in units of Λ) of Λ) 1 0.685-1.047 0.68-1.05 0.65 ± 0.05-1.05± 0.05 0.363 2 1.262-1.420 1.26-1.43 1.27 ± 0.01-1.45 ± 0.05 0.158 31.635-1.974 1.64-1.97 1.65 ± 0.05-2.05 ± 0.05 0.339 4 2.189-2.202 0.0135 2.624-2.779 0.155 6 3.322-3.405 0.083 7 3.619-3.679 0.059 83.893-3.934 0.071 9 4.271-4.402 0.131 10 5.239-5.400 0.161 116.218-6.244 0.026 12 6.914-6.916 0.0022 13 7.875-7.914 0.039 148.844-8.856 0.0113

To evaluate the accuracy of the foregoing quantitative predictions,numerical simulations of the surface modes of this same class of PBFswere conducted on a supercomputer using a full-vectorial plane waveexpansion method, as discussed above The dielectric was defined to besilica and the radius ρ of the air-holes 104 was defined to be equal to0.47Λ. The results of the simulations are plotted in FIG. 17 as opentriangles joined by dashes, wherein the right vertical axis representsthe number of surface modes predicted by the numerical simulations. Notethat this curve of triangular points is exactly the same as the curve oftriangular points of FIG. 9. The agreement with the predictions of thegeometric criterion (plotted as a solid curve in FIG. 17) is excellent.This agreement is further apparent by comparing the information in thesecond column of Table 1 for the boundary values of the first threesurface-mode-free bands generated by the geometric criterion with theinformation in the fourth column of Table 1 for the boundary valuesproduced by the simulations. The geometric criterion produces valuesthat are within 5% of the values produced by the simulations. Note thatthe exact boundary radii produced by the simulations were computed inlimited numbers (e.g., for the radii encompassing the first threesurface-mode-free bands) and were computed with a limited number ofdigits because the simulations are very time consuming (e.g., about sixhours per radius). In contrast, the geometric criterion provided farmore information in a small amount of time. Also note that although thegeometric criterion does not accurately predict the exact number ofsurface modes (see FIG. 17), the geometric criterion does exhibit thecorrect trend. In particular, the geometric criterion predicts thatsurface modes generally become more numerous with increasing radius R ofthe core 106, which is consistent with the original hypothesis.

The effect of the fiber air-filling ratio on the presence of surfacemodes can also be quickly evaluated with the above-described geometriccriterion by simply recalculating the boundary radii for differentvalues of the hole radius ρ. The results of the calculations areillustrated in FIG. 18, which plots the normalized boundary core radiusR/Λ, from R/Λ=0.6 to R/Λ=2.0, on the vertical axis versus the normalizedhole radius ρ/Λ, from ρ/Λ=0.43 to ρ/Λ=0.50, on the horizontal axis. Thepossible values for ρ are constrained between approximately 0.43Λ, belowwhich the photonic crystal has no bandgap, and below approximately0.50Λ, at which the thickness of the membranes 112 becomes zero. Theranges of core radii versus hole radii that support surface modes areshaded (cross hatched) and the ranges of core radii that do not supportsurface modes are unshaded. FIG. 18 shows that larger holes 104, whichhave greater air-filling ratios, yield wider surface-mode-free bandsbecause increasing the radius ρ of the air-holes 110 decreases theradius a of the rods (represented by the inscribed circles 114). Becauseof the smaller rod size, the ranges of core radii R that intersect therods are narrower, and the bands of surface-mode-free radii becomewider.

Other interesting observations can be obtained from the results of thestudies described above. First, in experimental PBFs 100, the core 106is typically created by removing the central seven tubes or the centralnineteen tubes from the preform. These configurations correspond to coreradii R of approximately 1.15Λ and approximately 2.1Λ, respectively. Thegeometric criterion defined herein confirms the predictions of exactsimulations that both of these configurations exhibit surface modes, asshown, for example, in FIG. 17. The existence of the surface modesexplains, at least in part, the high propagation loss of mostphotonic-bandgap fibers fabricated to date.

Second, the simulated curve in FIG. 17 shows that a small change in coreradius is all it takes to go from a surface-mode-free PBF to a PBF thatsupports surface modes. The abruptness of the transitions is consistentwith the perturbation process that creates surface modes, and supportsthe credibility of the rod approximation discussed above.

Third, the trends in Table 1 discussed earlier can be explained withsimple physical arguments. As the core radius increases, adjacentconcentric layers of rods become closer to each other, as shown in FIG.16. For larger radii, it is increasingly more difficult to find room fora circular radius that avoids all rods. Also, a larger radius tends tointersect more rods, and thus the number of surface modes generallyincreases. A manifestation of this effect can readily be seen in thefifth and sixth layers of rods, which lie between band 4 and band 5 inFIG. 16. The fifth and sixth layers overlap radially and thus merge intoa single, wider zone of core radii that support surface modes. In otherwords, there is no surface-mode-free band between the fifth and sixthlayers of rods. The same effect occurs with respect to the seventh,eighth and ninth layers, which lie between band 5 and band 6 in FIG. 16and cause the large numerical difference between the maximum radius ofband 5 (R=2.779Λ) and the minimum radius of band 6 (R=3.322Λ) inTable 1. Conversely, as the radius R of the core 106 increases, thesurface-mode-free bands become increasingly narrower, as can readily beseen in the fifth column of Table 1, which lists the width of eachsurface-mode-free band in units of Λ.

It can be expected intuitively that cores 106 with radii larger thansome critical value R_(C) will all support surface modes, and thus, onlya finite number of surface-mode-free bands are available. This intuitiveexpectation is consistent with the results of Table 1. In particular,for the structure evaluated herein for a radius ρ of the holes 104 of0.47Λ, the number of surface-mode-free bands is limited (i.e., only 14bands), and a critical radius R_(C) (i.e., approximately 8.86Λ) existsabove which the surface modes form a continuum. As indicated by thevalues in Table 1, the last four surface-mode-free bands are so narrow(e.g., ΔR of a few percent of Λ) that the last four bands are probablyunusable for most practical applications. A corollary of thisobservation is that multimode PBFs with the particular geometryillustrated herein and with a core radius R greater than 5.4Λ willlikely be plagued with surface modes.

The average value of the 1/e² radius of any of the lobes of the actualbulk mode in FIGS. 15A and 15B is approximately 0.22Λ. In comparison tothe intensity lobe, the radius a of the inscribed (dashed) circle inFIG. 8 is approximately 0.107Λ. A more refined figure and a betterquantitative agreement can be obtained by refining the value of theequivalent radius a of the silica rod, and by calculating the averageradius of the fundamental mode of a solid rod suspended in air.

Another observation obtained from the study described herein is thatsurface modes can be avoided in principle for any core size by selectinga non-circular core shape having a surface that does not intersect anyrods. A schematic of an example of a non-circular core having acharacteristic dimension corresponding to the shortest distance from thecenter to the nearest boundary of the core is shown in FIG. 19. With ahexagon-shaped core (as outlined by a dashed line in FIG. 19 to assistin visualizing the shape of the core), the introduction of any surfacemode is avoided even when the core region is large. Such a structurecould represent an improvement over the above-described circular corestructures in applications where multi-mode operation is desired.

The geometric criterion described herein is not limited to theparticular triangular geometry with circular cladding holes and thecircular cores. It is applicable to other shapes and geometries.

In accordance with the foregoing description, a simple geometriccriterion quickly evaluates whether an air-core PBF exhibits surfacemodes. Comparison of the results of the geometric criterion to theresults of numerical simulations demonstrates that when applied tofibers with a triangular-pattern cladding and a circular core, thegeometric criterion accurately predicts the presence of a finite numberof bands of core radii that support no surface modes. For sufficientlylarge circular cores (i.e., for radii above the largest of these bands),the fiber supports surface modes for any core radius. This versatilecriterion provides an expedient new tool to analyze the existence ofsurface modes in photonic-crystal fibers with an arbitrary crystalstructure and an arbitrary core profile.

FIGS. 20A and 20B illustrate plots of the effective refractive indicesof the modes as a function of wavelength. The plot in FIG. 20Aillustrates indices of the fiber manufactured by Crystal Fibre. The plotin FIG. 20B illustrates the indices of the fiber manufactured byCorning. The plots were generated using numerical simulations. Thefundamental core modes are shown in bold curves, and the less intenselines are the surface modes. The Crystal Fibre core mode (FIG. 20A) hasa measured minimum loss of the order of 100 dB/km while the Corning coremode (FIG. 20B) has a measured minimum loss of 13 dB/km. The loss of thecore mode is believed to be mainly due to coupling of the core mode tosurface modes, which are inherently lossy due to the concentration ofenergy near the surface of the core. Hence surface modes suffer fromenhanced Rayleigh scattering. The total power coupled from core modes tosurface modes will be enhanced, and thus the loss will be larger, if thecore supports a large number of surface modes. In addition, it is wellknown from coupled mode theory that the coupling of two modes, in thiscase the core mode to a surface mode, will be stronger when theeffective refractive indexes of the two modes are closer.

When considering the modes at a wavelength of 1.50 μm in FIGS. 20A and20B, it can be seen that there are far more surface modes in the CrystalFibre structure (FIG. 20A) than in the Corning structure (FIG. 20B).Furthermore, the effective refractive indices of the Corning surfacemodes are less than 0.986, while the core mode has an effectiverefractive index of 0.994, a 0.8% difference. On the other hand, thecore mode in the Crystal Fibre structure has an effective refractiveindex of 0.996, while the nearest surface mode has an effectiverefractive index of 0.994, only a 0.2% difference. Everything else beingthe same, in particular the level of geometrical perturbation present inthe core of the two fibers, coupling of the core mode to surface modesis expected to be stronger in the fiber manufactured by Crystal Fibre.Thus, the Crystal Fibre fiber supports more surface modes, and thesurface modes couple more strongly, which is consistent with the higherpropagation loss of the Crystal Fibre fiber. From the foregoing, it canbe concluded that to design air-guided PBFs with a low loss, thepreferred approach is to completely eliminate surface modes, asdescribed above. If it is not possible to completely eliminate thesurface modes, a second approach is to reduce the number of surfacemodes (e.g., by assuring that the core does not cut through too manycorners of the cladding lattice), to increase the effective indexdetuning between the core modes and the remaining surface modes, orboth.

Photonic-Bandgap Fibers with Core Rings

The discussion above describes a detailed investigation of the existenceof surface modes in particular types of PBFs 100. An example of a PBFthat does not support surface modes is illustrated in FIG. 21A. Thefiber cladding comprises a photonic crystal lattice 102 with holes 104(e.g., holes filled with air) having substantially circularcross-sections and arranged in a triangular pattern in silica. The holes104 have a period Λ, and each hole 104 has a hole radius ρ.

The fiber core 106 of FIG. 21A comprises an air-hole with a generallycircular cross-section with a radius R and which is centered on one ofthe holes 104, as if a hole of radius R had been drilled into the fiber.As described above, simulations and physical explanations show that ifthe surface of the core 106 cuts through the thicker portions of thesilica lattice 102 between three surrounding holes 104 (which arereferred to as corners 110 in FIG. 21A), the core 106 will supportsurface modes. But if the surface of the core 106 intersects only thethinner portions of the lattice 102 between only two adjacent holes 104(referred to as membranes 112 in FIG. 21A), the core 106 will besubstantially free of surface modes. As described above, this criterioncan be used to design air-core PBFs 100 that do not substantiallysupport surface modes at any frequency in the bandgap and thuspresumably fibers 100 that exhibit significantly lower losses.

FIG. 21B illustrates an exemplary embodiment of a PBF 200 having thesame general geometry as the PBF 100 of FIG. 21A. Besides comprising alattice 202, a plurality of holes 204, and a core 206, the PBF 200further comprises a core ring 220 surrounding the core 206. In certainembodiments, the core ring 220 extends along the PBF 200 and has across-sectional shape which generally surrounds the core 206. In certainembodiments, the ring 220 has an inner perimeter and an outer perimeterwhich are generally parallel to one another along the length of the PBF100, along an azimuthal direction around an axis of the PBF 100, orboth. In certain other embodiments, the inner perimeter and the outerperimeter are not parallel to one another along the length of the PBF100, along an azimuthal direction around an axis of the PBF 100, orboth. In certain embodiments, the core ring 220 has a thickness which isgenerally constant along the length of the PBF 100, along an azimuthaldirection around an axis of the PBF 100, or both. In certainembodiments, the core ring 220 has a thickness which varies along thelength of the PBF 100, along an azimuthal direction around an axis ofthe PBF 100, or both.

The lattice 202 of FIG. 21B has corners 210 and membranes 212, asdescribed above. As described more fully below, computer simulations ofa PBF 200 having a thin core ring 220 provide information regarding theeffects of the thin core ring 220 surrounding the core 206 of the PBF200 and regarding the existence of surface modes. These simulations showthat the addition of even a very thin silica ring 220 (e.g., a thicknessof approximately 0.03Λ) introduces surface modes. These surface modeshave main maxima centered on the open segments of the ring 220, and thedispersion curves of these surface modes lie between the upper band edgeof the photonic bandgap and the light line. These features indicate thatthese surface modes are the guided modes of the ring 220 surrounded byair within an inner perimeter 222 of the ring 220 and the photoniccrystal lattice 202 outside an outer perimeter 224 of the ring 220. Thepresence of a ring 220 also induces small distortions and decreasedgroup velocity dispersion in the fundamental core mode, and also inducesa small frequency down-shift. The intensity distributions of the ringsurface modes suggest that the ring surface modes also introducesubstantial losses to the core mode, a postulate which is stronglysupported by published experimental evidence. The propagation loss ofair-core fibers may be reduced by various techniques, including but notlimited to, (i) fabricating fibers without a core ring (as describedabove); (ii) keeping the ring 220 but selecting the radius of the ring220 towards the upper end of the single-mode range (e.g., R<˜1.2Λ) toincrease the detuning between the core 206 and the surface modes; (iii)reducing the ring thickness to reduce the number of ring surface modes;or (iv) a combination of these actions.

In certain embodiments, a photonic bandgap fiber (PBF) has a claddingphotonic crystal region comprising a triangular lattice composed of aplurality of holes in silica, where the holes have substantiallycircular cross-sections and are spaced apart by a period A. Certain suchPBFs, in which the holes are filled with air, are described, forexample, in R. F. Cregan et al., Single-Mode Photonic Band Gap Guidanceof Light in Air, Science, Vol. 285, 3 Sep. 1999, pages 1537-1539; JesBroeng et al., Analysis of air guiding photonic bandgap fibers, OpticsLetters, Vol. 25, No. 2, Jan. 15, 2000, pages 96-98; and Jes Broeng etal., Photonic Crystal Fibers: A New Class of Optical Waveguides, OpticalFiber Technology, Vol. 5, 1999, pages 305-330, which are herebyincorporated herein by reference.

In practice, the cross-sectional profile of an air-core fiber issomewhat different from the cross-sectional profile shown in FIG. 21A.The PBF of certain embodiments is drawn from a preform made of silicacapillary tubes stacked in a hexagonal arrangement, and a few tubes areremoved from the center of the stack to form the core. For example, toproduce a typical single-mode core, seven tubes are removed, asillustrated in FIG. 22. During the process of drawing of a fiber fromthe preform, surface tension pulls on the softened glass walls of thetubes to cause the original scalloped outline of the core shown in FIG.22 to become a smooth thin ring of silica. Such a core ring is astandard feature of current commercial air-core PBFs, as described, forexample, in Dirk Müller et al., Measurement of Photonic Band-gap FiberTransmission from 1.0 to 3.0 μm and Impact of Surface Mode Coupling,Proceedings of Conference on Laser and Electro-Optics (CLEO) 2003,Baltimore, USA, 1-6 Jun. 2003, paper QTuL2, 2 pages; B. J. Mangan etal., Low loss (1.7 dB/km) hollow core photonic bandgap fiber, Conferenceon Optical Fiber Communications, OFC 2004, Los Angeles, Calif., Feb.22-27, 2004, Postdeadline Paper PDP24, 3 pages; and Theis P. Hansen etal., Air-Guiding Photonic Bandgap Fibers. Spectral Properties,Macrobending Loss, and Practical Handling, IEEE Journal of LightwaveTechnology, Vol. 22, No. 1, January 2004, pages 11-15, which are herebyincorporated herein by reference.

The presence of a ring at the edge of the core introduces new boundaryconditions that did not exist in the fiber geometry discussed above withregard to the photonic-bandgap fiber with no core ring. Consequently,new sets of surface modes are expected to be present in a ringedair-core fiber. As described below, computer simulations confirm that acore ring does introduce surface modes, even when the ring is relativelythin. In certain embodiments, the ring thickness for the air-core PBF isless than 0.03Λ, where Λ is the period of the crystal, while in otherembodiments, the ring thickness is less than 0.02Λ, and in still otherembodiments, the ring thickness is less than 0.01Λ. The surface modesare found to be the guided modes of the ring itself. The ring issurrounded by the material filling the core (e.g., air) on its insideand by the photonic crystal on its outside and acts as a waveguide. Thering also induces small but noticeable perturbations of the fundamentalcore mode, including intensity profile distortions, increasedgroup-velocity dispersion, and frequency down-shift. For certainphotonic-bandgap fibers, these ring surface modes can introducesubstantial propagation loss of core modes. As described herein,computer simulations advantageously provide information to betterunderstand the behavior of these surface modes and to configure thephotonic-bandgap fiber to reduce or eliminate the surface modes in orderto further reduce the losses of the air-core fibers.

In certain embodiments, a photonic-bandgap fiber 200 comprises aphotonic crystal lattice 202 comprising a first material (e.g., silica)with holes 204 filled with a second material (e.g., air) having arefractive index smaller than that of the first material. In certainembodiments, the holes 204 each have a radius ρ=0.47Λ and are arrangedin a triangular pattern, as illustrated in FIG. 21B. A larger holecomprising the second material and having a radius R is added to thisstructure to break its symmetry and to form the central core 206. In afirst set of simulations, the effects of a thin silica ring 220 at theperiphery of the core 206 are investigated.

In FIGS. 21A and 21B, the core radius is R=0.9Λ. In certain embodiments,this particular value for the core radius is selected so that thesurface of the core 206 cuts only through membranes 212 of the claddinglattice 202, as is readily apparent in FIG. 21B. As described above, inthe absence of the thin silica ring 220, the PBF of certain embodimentssupports no surface modes. The ring 220 in FIG. 21B has an inner radiusR₁=0.9Λ and has a small thickness of 0.03Λ. Thus, the ring 220illustrated by FIG. 21B has an outer radius R₇=0.93Λ.

The bulk modes, core modes, and surface modes of the fibers 100, 200shown in FIGS. 21A and 21B were calculated numerically on asupercomputer using a full-vectorial plane wave expansion method. See,for example, Steven G. Johnson et al., Block-iterative frequency-domainmethods for Maxwell's equations in a planewave basis, Optics Express,Vol. 8, No. 3, 29 Jan. 2001, pages 173-190, which is hereby incorporatedherein by reference. A supercell size of 10×10 and a grid resolution ofΛ/16 were used for the calculations.

FIG. 23A illustrates a calculated ω-k diagram of the fiber 100 when thecore 106 is not surrounded by a ring (e.g., the fiber 100 shown in FIG.21A). The dashed curves in FIG. 23A represent the edges of the fiberbandgap. As described above, for the radius R=0.9Λ, the core 106supports only the fundamental mode (which is in fact two-fold degeneratein polarization).

When the thin core ring 220 is added as shown in FIG. 21B, two mainchanges occur in the dispersion diagram, which are shown in FIG. 23B.First, as illustrated by a solid curve in FIG. 23B, the dispersion curveof the fundamental mode shifts towards lower frequencies. By comparingthe dispersion curve with the ring 220 of FIG. 23B with the dispersioncurve without the ring of FIG. 23A (reproduced as a dashed curve in FIG.23B), the shift towards lower frequencies is apparent.

The intensity profiles of the fundamental modes of the fiber 100calculated without the ring are plotted in FIG. 24A. The intensityprofiles of the fundamental modes of the fiber 200 with the ring 220 areplotted in FIG. 24B. Without a ring (FIG. 24A), the fundamental mode isstrongly localized to the core region. When the ring 220 is added (FIG.24B), the fundamental mode exhibits radial ridges with a six-foldsymmetry. The ridges are caused by the ring 220 having a higherrefractive index than the air in the core 206, so that the ring 220 actsas a local guide and pulls some of the mode energy away from the centerof the core 206. Because a slightly higher fraction of the mode energyis now contained in the silica ring 220, the mode is slowed down alittle, which explains the down-shift in its dispersion curve shown inFIG. 23B. The group velocity of the fundamental mode is also decreased.

The second change in the dispersion curve shown in FIG. 23B is theappearance of five new modes (three non-degenerate and two degenerate,shown as dotted curves in FIG. 23B). The intensity profiles of two ofthese modes illustrated in FIGS. 25A and 25B show that they are surfacemodes. Both intensity profiles in FIGS. 25A and 25B exhibit narrowmaxima centered on open segments of the ring which decay sharply intothe air on both sides of these segments. (As used herein, an “opensegment” of the ring is a portion of the ring spanning between twomembrane portions cut by the formation of the core.) The existence ofsuch surface modes has been reported before by in K. Saitoh et al.,Air-core photonic band-gap fibers: the impact of surface modes, which iscited above. Unlike the surface modes introduced by an air core, thesemodes do not result from perturbation of the photonic crystal bulkmodes. Rather, the modes result from the introduction of the thin ringof high index material in air, which acts as a waveguide. If the modeswere truly the modes of a ring surrounded on both sides by nothing butair, they would be slower than light; however, FIG. 23B shows that thisis not the case—the dispersion curves of all these modes are above thelight line. The photonic crystal on the outside of the ring speeds upthese modes to above the speed of light for the same physical reasonsthat cause the phase velocity of the core modes to be greater than thespeed of light. Note also that these surface modes introduced by thethin ring differ from the surface modes introduced by the air core alonein the sense that they do not intersect the light line within thebandgap.

Careful inspection of FIGS. 25A and 25B shows that the maxima of thesurface modes are not centered exactly on the ring but are located inthe air core just inside the ring. This is not the physical location ofthe maxima. Rather, the offset of the maxima is an artifact resultingfrom a limitation in the simulator. The simulator uses Fouriertransforms to calculate the fiber modes and automatically smoothens theoriginally abrupt edges of the photonic crystal dielectric to avoidintroducing unphysical oscillations in the solutions it generates. Thus,the simulator models a ring with a refractive index that graduallytapers from the index of silica down to the index of air over somedistance on both sides of the ring. As a result, the modeled ring is alittle thicker than the ring shown in FIGS. 25A and 25B. This offset ofthe maxima could be avoided in other calculations by increasing the gridresolution, which would use considerably more memory and would require amuch longer computing time. This artifact, however, does not affect thequantitative conclusions of these simulations. The artifact onlyproduces an uncertainty for the actual thickness of the modeled ring,which is somewhat larger than the nominal value of 0.03Λ initiallyselected.

In certain embodiments, the dimensions of the core ring are selected toreduce losses of the photonic-bandgap fiber and the amount of couplingbetween the fundamental modes of the fiber and the surface modes inducedby the core or by the core ring. As used herein, “core-induced surfacemodes” denotes modes which result from the existence of the core withinthe fiber, and “ring-induced surface modes” denotes modes which resultfrom the existence of the core ring within the fiber. In certainembodiments, the dimensions of the core ring are selected to reduce thenumber of core-induced surface modes, the number of ring-induced surfacemodes, or both. In embodiments in which the core ring has an outerperimeter, an inner perimeter, and a thickness between the outerperimeter and the inner perimeter, at least one of the outer perimeter,the inner perimeter, and the thickness is selected to reduce the numberof core-induced surface modes, the number of ring-induced surface modes,or both. For certain embodiments comprising a core ring having asubstantially circular cross-section, at least one of the outer radius,the inner radius, and the thickness is selected to reduce the number ofcore-induced surface modes, the number of ring-induced surface modes, orboth.

In certain embodiments, the radius of the core region (corresponding tothe outer radius of the core ring) is selected to minimize the number ofcore-induced surface modes. As described above, by having an outerperimeter which passes only through regions of the photonic crystallattice which do not support intensity lobes of the highest frequencybulk mode, certain embodiments described herein substantially avoidcreating core-induced surface modes. Exemplary ranges or values of theouter radius of the core ring of a single-mode fiber compatible withcertain such embodiments include, but are not limited to, less than1.2Λ, between approximately 0.9Λ and approximately 1.13Λ, betweenapproximately 0.7Λ and approximately 1.05Λ, and approximately equal to0.8Λ. Exemplary ranges or values of the outer radius of the core ring ofa multi-mode fiber compatible with certain such embodiments include, butare not limited to, between approximately 1.25Λ and approximately 1.4Λ,between approximately 1.6Λ and approximately 2.0Λ, between approximately2.1Λ and approximately 2.2Λ, between approximately 2.6Λ andapproximately 2.8Λ, and between approximately 3.3Λ and approximately3.4Λ.

In certain embodiments, the thickness of the core ring is selected tominimize the number of ring-induced surface modes. As described morefully below, thinner core rings generally support fewer ring-inducedsurface modes than do thicker core rings. In certain embodiments, thering thickness is selected to be sufficiently small to support at mostone ring-induced surface mode. Exemplary ranges or values of the ringthickness compatible with certain embodiments described herein include,but are not limited to, less than 0.03Λ, less than 0.02Λ, and less than0.01Λ.

A second set of simulations models the effects of adding a thin ring toan air-core fiber that already supports surface modes. To do so, thesimulation uses the same fiber as before, except that the core radius isincreased to R₁=1.13Λ. The cross-sectional profile of this modifiedfiber is shown in FIG. 26A. The fiber core now cuts through corners ofthe dielectric lattice. As a consequence, surface modes are now present,even in the absence of a ring, as described above.

The calculated ω-k diagram of the fiber of FIG. 26A is plotted in FIG.27A. As predicted, in addition to the two degenerate core modes, thefiber of FIG. 26A exhibits several surface modes (six in this case, twodegenerate and four non-degenerate). The origin of these surface modeshas been discussed above in relation to the photonic-bandgap fibers withno core rings. In short, when the air core is introduced in aphotonic-crystal cladding, the core terminates the dielectric latticeabruptly all around the edges of the core. The core locally replacesdielectric material with air, and all the original bulk modes of thefiber are perturbed. The portions of the core surface that cut throughthe air holes in FIG. 21A can be considered to have replaced the airholes with air of the core, and thus induce a comparatively weakperturbation. See, for example, F. Ramos-Mendieta et al., Surfaceelectromagnetic waves in two-dimensional photonic crystals: effect ofthe position of the surface plane, Physical Review B, Vol. 59, No. 23,June 1999, pages 15112-15120, which is hereby incorporated herein byreference. In contrast, the portions of the core surface that cutthrough the dielectric regions of the photonic crystal in FIG. 21Areplace the dielectric material of the lattice with air of the core, andthus perturbs the bulk modes more strongly.

Whether surface modes are induced by the introduction of the coredepends on the magnitude of the perturbation of the bulk modes, and themagnitude of the perturbation of the bulk modes depends on which areasof the dielectric regions are intersected by the core. As describedabove in relation to photonic-bandgap fibers with no core ring, if thecore radius is such that the core surface only cuts through dielectricmembranes, which are relatively thin, the perturbation is not strongenough to induce surface modes, but when the core surface cuts throughone or more dielectric corners, the perturbation is stronger and surfacemodes are induced.

The foregoing description explains why surface modes are present in thefiber of FIG. 26A, but are not present in the fiber of FIG. 21A. Becausethe index perturbation is negative (e.g., higher index material isreplaced by lower index material), the bulk modes in the ω-k diagram allshift down in the k_(z) space from their respective unperturbedposition, or equivalently the bulk modes all shift up in frequency. Fora silica/air PBF, the index difference is small enough that this shiftis small. Almost all perturbed bulk modes thus remain in a bulk-modeband and do not induce surface modes. The exceptions are the modes inthe highest frequency bulk-mode band of the lower band. Because suchmodes are located just below the bandgap in the ω-k diagram, theperturbation moves such modes into the bandgap as surface modes. See,for example, F. Ramos-Mendieta et al., Surface electromagnetic waves intwo-dimensional photonic crystals: effect of the position of the surfaceplane, which is cited above. These surface modes expectedly have thesame symmetry as the bulk modes from which they originate, e.g., thesurface modes all exhibit narrow lobes centered on dielectric corners ofthe lattice. See, for example, Michel J. F. Digonnet et al., Simplegeometric criterion to predict the existence of surface modes inair-core photonic-bandgap fibers, which is cited above. This is true ofall the surface modes of FIG. 27A. Parenthetically, the dispersioncurves of the surface modes all cross the light line within the bandgap.

When a thin silica ring is added around the core of this fiber as shownin FIG. 26B, the ω-k diagram evolves to the new diagram shown in FIG.27B. Again, the two nearly degenerate fundamental modes are slightlyslowed down. Thus, the fundamental modes are shifted down in frequency.For the same physical reasons, the six pre-existing surface modes arealso frequency down-shifted. This shift is larger for the surface modesthan for the fundamental modes, which means that the surface modes arepushed away from the fundamental modes. Thus, in certain embodiments,the introduction of a thin core ring advantageously decreases thecoupling efficiency of pre-existing surface modes to the fundamentalmodes, which in turn decreases the fiber loss.

In certain embodiments, the addition of the thin ring also introduces anew group of surface modes, as can be seen in FIG. 27B, whichdeteriorates the fiber loss. These surface modes (two degenerate andthree non-degenerate, as before) are once again supported by the opensegments of the ring and are sped up by the presence of the photoniccrystal. As expected, the profiles of the introduced surface modes inFIG. 27B are qualitatively similar to the profiles of the surface modesintroduced by a ring in an air-core fiber that did not originallysupport surface modes, as shown, for example, in FIGS. 25A and 25B.

Since the addition of a ring replaces air by dielectric material (e.g.,silica) in the air core, the perturbation introduced by the ring has theopposite sign as when the air core alone is introduced. The indexperturbation is now negative (e.g., lower index material is replaced byhigher index material), and in the ω-k diagram, all the bulk modes shiftdown in frequency. In embodiments in which this perturbation issufficiently large, the lowest frequency bulk modes of the upper bandshift down into the bandgap and shed surface modes. This result is themirror image of what happens when an air core alone is introduced in thePBF, in which case the index perturbation is positive and the highestfrequency bulk modes of the lower band move into the bandgap as surfacemodes. In the ω-k diagram, the dispersion curves of the surface modesshed by the upper-band modes are located just below the upper band fromwhich they originate, e.g., in the same general location as thering-supported surface modes shown in FIG. 27B. However, such surfacemodes would also have the symmetry of upper-band bulk modes, e.g., theirmaxima would be on the membranes of the photonic crystal lattice.Because none of the surface modes in FIG. 27B exhibit the expectedsymmetry, in certain such embodiments, the ring is thin enough that theperturbation it introduces on the lattice is too weak to generatesurface modes from the upper band.

Corning researchers have provided experimental evidence of the presenceof surface modes in some of their air-core fibers. See, for example,Dirk Müller et al., Measurement of Photonic Band-gap Fiber Transmissionfrom 1.0 to 3.0 μm and Impact of Surface Mode Coupling, which is citedabove. For a fiber having a thin hexagonal ring around the core, theCorning researchers measured a strong attenuation peak located around1600 nanometers, roughly in the middle of the fiber transmissionspectrum. They inferred through simulations that this peak was due to asmall number of surface modes that happen to cross the dispersion curveof the fundamental mode around 1600 nanometers. In this spectral region,coupling between the core mode and these surface modes was thereforeresonant, which resulted in the attenuation peak observed by theresearchers.

Using the published cross-sectional profile of the foregoing Corningfiber as input to the simulation code, simulations similar to thosedescribed herein confirmed the findings of the Corning researchers. Inaddition, these simulations showed that these resonant surface modeshave maxima centered on the open segments of the hexagonal core ring ofthe fiber, which implies that the surface modes are again entirelysupported by the ring.

Some experimental air-core PBFs with a ring around the core exhibit aspurious mid-band attenuation peak, as described for example, in theabove-cited Dirk Müller et al., Measurement of Photonic Band-gap FiberTransmission from 1.0 to 3.0 μm and Impact of Surface Mode Coupling, andB. J. Mangan et al., Low loss (1.7 dB/km) hollow core photonic bandgapfiber, which are cited above. Other experimental air-core PBFs do notexhibit this peak. See, for example, Theis P. Hansen et al., Air-GuidingPhotonic Bandgap Fibers: Spectral Properties, Macrobending Loss, andPractical Handling, which is cited above, and HC-1550-02 Hollow CorePhotonic Bandgap Fiber, blazephotonics.com, pages 1-4; which is herebyincorporated herein by reference. These differences indicate thatparticular combinations of ring and crystal geometries move the surfacemodes into the middle of the band while other combinations do not. Thus,in certain embodiments, the ring and crystal geometries areadvantageously designed to reduce or avoid the surface modes within theband.

In certain embodiments, even if the core radius of a ring-less PBF isselected to avoid the surface modes shed by the lower-band bulk modes,as described above in relation to photonic-bandgap fibers with no corering, surface modes are still likely to be present once a ring isintroduced. Because these surface modes have high electric fields in thedielectric ring, these surface modes are expected to be lossy, likeother surface modes. Furthermore, because the fundamental core mode hasrelatively large fields on the open segments of the ring, shown in FIG.24B, the overlap of the core mode with the surface modes supported bythe ring is larger than in a ring-less fiber. Thus, in certainembodiments, coupling from the core mode to these surface modes ispredicted to be larger than in a ring-less fiber, which means that thesesurface modes likely introduce a substantial loss and are thusundesirable. This view is fully supported by the simulations andexperimental observations regarding Corning's fiber discussed above.

The loss induced by ring surface modes can be alleviated in variousways. In certain embodiments, fibers are fabricated without a core ring,as described above. In certain such embodiments, the fiber carries asingle mode and no surface modes across the entire bandgap.

In certain other embodiments, one or more of the outer ring radius, theinner ring radius, and the ring thickness are selected to reduce lossesof the PBF. Certain such embodiments advantageously reduce the amount offiber loss as compared to other configurations by reducing the number ofring-induced surface modes, reducing the coupling between thering-induced surface modes and the fundamental mode (e.g., by moving thering-induced surface modes away from the fundamental mode in the ω-kdiagram), or both.

In certain embodiments, the core ring of the photonic-bandgap fibersurrounds the core and has an outer perimeter which passes throughregions of the photonic crystal lattice which support intensity lobes ofthe highest frequency bulk mode. In certain other embodiments, the corering surrounds the core and has an outer perimeter which passes throughone or more of the inscribed circles enclosed by a portion of thelattice material and having a circumference tangential to three adjacentholes.

FIG. 26B illustrates an exemplary embodiment in which the core ring hasa substantially circular cross-section, an inner perimeter generallyparallel to the outer perimeter, and a thickness between the innerperimeter and the outer perimeter. Other shapes of the core ring (e.g.,non-circular cross-sections, non-parallel inner and outer perimeters)are also compatible with embodiments described herein. In addition,while the fiber lattice of FIG. 26B has a triangular pattern of holeswith substantially circular cross-sections, other patterns and shapes ofholes are also compatible with embodiments described herein.

As described above, for a constant ring thickness (e.g., 0.03Λ), whenthe ring radius (e.g., an outer ring radius) is increased from 0.9Λ to1.13Λ the number of ring surface modes remains unchanged (five in bothcases). Thus, over the range of core radii for which the fiber issingle-moded (R<˜1.2Λ), the number of surface modes remains constant.However, increasing the ring radius from 0.9Λ to 1.13Λ shifts thedispersion curve of the ring surface modes up in frequency, e.g., awayfrom the dispersion curve of the fundamental mode, as can be seen bycomparing FIG. 23B and FIG. 27B. In certain embodiments, the outerradius of the core ring is sufficiently large to shift the dispersioncurves of the ring-induced surface modes away from the dispersion curveof the fundamental mode of the fiber. Using a large core radius incertain embodiments therefore advantageously increases the detuningbetween the fundamental and surface modes, and thus weakens the couplingbetween the fundamental mode and the surface modes and lowers the fiberloss. In certain embodiments, the outer radius of the core ring isselected to induce ring surface modes having dispersion curves which aresubstantially decoupled from the fundamental mode dispersion curve ofthe fiber.

For a fixed ring radius (R=0.9Λ), the number of ring surface modesincreases with increasing ring thickness, e.g., from five modes forthicknesses of 0.03Λ and 0.06Λ to ten modes for a thickness of 0.09Λ. Incertain embodiments, the core ring has a thickness selected to besufficiently small to support at most one ring-induced surface mode.Using a thinner ring in certain embodiments therefore advantageouslyreduces the number of surface modes, and thus reduces the fiber loss.Accurate modeling of PBFs with a ring substantially thinner than 0.03Λwould typically utilize a much lower grid resolution (e.g., <Λ/16) thanthat used in the simulations discussed herein, with a correspondingincrease of computation time and memory requirements.

FIG. 28 illustrates an alternative photonic crystal (PC) structure whichis terminated across its membranes by an infinite thin plane slab, andwhich is used to model a PBF. The PC has the same triangular holepattern in silica and has the same air hole radius (ρ=0.47Λ) as the PBFconfigurations discussed above. The advantage of using the photoniccrystal of FIG. 28 rather than a fiber structure is that the PC symmetryallows a much smaller supercell size to be used. For example, a 1×16√3supercell size can be used in this case instead of a 10×10 supercellsize. Consequently, for a given computation time, the grid resolutioncan be much smaller (˜Λ/64).

Simulations using the PC configuration of FIG. 28 show that the thinslab supports surface modes centered on the open segments of the slab,as expected. The number of slab surface modes was found to decrease withdecreasing slab thickness. As the slab thickness is reduced towards zero(0.001Λ is the lowest value modeled), the number of surface modessupported by the slab may drop to zero or to a non-zero asymptoticvalue. In addition, the surface mode shifts into the bandgap as the ringthickness is reduced. In certain embodiments, the ring can supportsurface modes even for vanishingly small thicknesses.

Various embodiments have been described above. Although this inventionhas been described with reference to these specific embodiments, thedescriptions are intended to be illustrative of the invention and arenot intended to be limiting. Various modifications and applications mayoccur to those skilled in the art without departing from the true spiritand scope of the invention as defined in the appended claims.

1.-20. (canceled)
 21. An optical fiber comprising: a cladding comprisinga material having a first refractive index and a pattern of regionsformed therein, the regions having a second refractive index lower thanthe first refractive index, wherein the optical fiber supportspropagation of a highest frequency bulk mode comprising intensity lobeshaving maxima located within the cladding; and a core regionsubstantially surrounded by the cladding, the core region having anouter perimeter that does not intersect the maxima of the intensitylobes.
 22. The optical fiber of claim 21, wherein the pattern isperiodic.
 23. The optical fiber of claim 21, wherein the regions in thematerial are circular, the pattern of regions is arranged such that eachgroup of three adjacent regions forms a triangle with a respective firstportion of the material between each pair of regions and with arespective second portion of the material forming a central area withineach group of three adjacent regions, and the outer perimeter of thecore region passes only through the first portions of the material. 24.The optical fiber of claim 23, wherein the regions in the material areholes having walls defined by the surrounding material.
 25. The opticalfiber of claim 24, wherein the holes in the material are hollow.
 26. Theoptical fiber of claim 25, wherein the holes in the material are filledwith a gas having the second refractive index.
 27. The optical fiber ofclaim 21, wherein the material is a dielectric.
 28. The optical fiber ofclaim 21, wherein the material is silica.
 29. The optical fiber of claim21, wherein the core region comprises a core ring having an innerperimeter, an outer perimeter, and a thickness between the innerperimeter and the outer perimeter.
 30. The optical fiber of claim 29,wherein the thickness is sized to reduce the number of ring surfacemodes supported by the core ring.
 31. The optical fiber of claim 29,wherein the core ring generally surrounds the core region and thecladding generally surrounds the core ring.
 32. The optical fiber ofclaim 29, wherein the core ring comprises the material.
 33. The opticalfiber of claim 29, wherein the pattern of regions comprises a pluralityof geometric regions, each geometric region having a cross-section witha respective center and adjacent geometric regions being spaced apart bya center-to-center distance Λ.
 34. The optical fiber of claim 33,wherein the thickness is less than approximately 0.03Λ.
 35. The opticalfiber of claim 33, wherein the core ring has a substantially circularcross-section with an outer radius less than 1.2Λ.
 36. The optical fiberof claim 33, wherein the core ring has a substantially circularcross-section with an outer radius between approximately 1.25Λ andapproximately 1.4Λ.
 37. The optical fiber of claim 33, wherein the corering has a substantially circular cross-section with an outer radiusbetween approximately 1.6Λ and approximately 2.0Λ.
 38. A method ofpropagating light signals, the method comprising: providing an opticalfiber comprising a cladding comprising a material having a firstrefractive index and a pattern of regions formed therein, the regionshaving a second refractive index lower than the first refractive index,wherein the optical fiber supports propagation of a highest frequencybulk mode comprising intensity lobes having maxima located within thecladding, the optical fiber further comprising a core regionsubstantially surrounded by the cladding, the core region having anouter perimeter that does not intersect the maxima of the intensitylobes; and propagating light through the optical fiber.
 39. A method ofmaking an optical fiber, the optical fiber comprising a material havinga first refractive index and a pattern of regions formed therein, theregions having a second refractive index lower than the first refractiveindex, the method comprising: determining locations of intensity lobesof a highest frequency bulk mode within the cladding; and forming a coreregion having an outer perimeter that does not intersect the maxima ofthe intensity lobes.